(1) Whenever two or more counties establish a facility or facilities as provided by section 23-3515, the county board of each such county shall proceed immediately to appoint three members to serve on the board of trustees of such facility or facilities. When such board is first established, each county board shall appoint one member for a term of two years, one for four years, and one for six years from the date they are appointed. Thereafter, as their terms expire, members shall be appointed for a term of six years.
(2) Whenever the board would consist of an even number of members, one additional member shall be appointed for a term of six years by the county board of the county having the greatest population as disclosed by the latest United States census.
(3) The board of trustees provided for in this section shall have the same powers, duties, obligations, and authority provided in sections 23-3502 to 23-3509.